Definition of the stress tensor

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 76-77

Section 1.3.2.2. Definition of the stress tensor

A. Authier^a ^* and A. Zarembowitch^b

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and ^bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence e-mail: aauthier@wanadoo.fr

1.3.2.2. Definition of the stress tensor

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Using the condition on the resultant of forces, it is possible to show that the components of the stress $[{\bf T}_{n}]$ can be determined from the knowledge of the orientation of the normal n and of the components of a rank-two tensor. Let P be a point situated inside volume V, $[Px_{1}]$ , $[Px_{2}]$ and $[Px_{3}]$ three orthonormal axes, and consider a plane of arbitrary orientation that cuts the three axes at Q, R and S, respectively (Fig. 1.3.2.3). The small volume element PQRS is limited by four surfaces to which stresses are applied. The normals to the surfaces PRS, PSQ and PQR will be assumed to be directed towards the interior of the small volume. By contrast, for reasons that will become apparent later, the normal n applied to the surface QRS will be oriented towards the exterior. The corresponding applied forces are thus given in Table 1.3.2.1. The volume PQRS is subjected to five forces: the forces applied to each surface and the resultant of the volume forces and the inertial forces. The equilibrium of the small volume requires that the resultant of these forces be equal to zero and one can write $[-{\bf T}_{n} \hbox{ d}\sigma + {\bf T}_{1} \hbox{ d}\sigma_{1} + {\bf T}_{2}\hbox{ d}\sigma_{2} + {\bf T}_{3}\hbox{ d}\sigma_{3} + {\bf F}\rho \hbox{ d}\tau = 0 ]$ (including the inertial forces in the volume forces).

Table 1.3.2.1 | top | pdf |
Stresses applied to the faces surrounding a volume element

$[\alpha_{1}]$ , $[\alpha_{2}]$ and $[\alpha_{3}]$ are the direction cosines of the normal n to the small surface QRS.

Face	Area	Applied stress	Applied force
QRS	dσ	$[-{\bf T}_{n}]$	$[-{\bf T}_{n}\hbox{ d}\sigma ]$
PRS	$[\hbox{d}\sigma_{1} = \alpha_{1} \hbox{ d}\sigma ]$	$[{\bf T}_{1}]$	$[{\bf T}_{n} \hbox{ d}\sigma_{1} ]$
PSQ	$[\hbox{d}\sigma_{2} = \alpha_{2} \hbox{ d}\sigma ]$	$[{\bf T}_{2}]$	$[{\bf T}_{n} \hbox{ d}\sigma_{2} ]$
PQR	$[\hbox{d}\sigma_{3} = \alpha_{3} \hbox{ d}\sigma ]$	$[{\bf T}_{3}]$	$[{\bf T}_{n} \hbox{ d}\sigma_{3} ]$

Figure 1.3.2.3 | top | pdf |

Equilibrium of a small volume element.

As long as the surface element dσ is finite, however small, it is possible to divide both terms of the equation by it. If one introduces the direction cosines, $[\alpha_{i}]$ , the equation becomes $[-{\bf T}_{n} + {\bf T}_{1}\hbox{ d}\alpha_{1} + {\bf T}_{2}\hbox{ d}\alpha_{2} + {\bf T}_{3}\hbox{ d}\alpha_{3} + {\bf F}\rho \hbox{ d}\tau/\hbox{d}\sigma= 0. ]$ When dσ tends to zero, the ratio $[\hbox{d}\sigma/\hbox{d}\tau ]$ tends towards zero at the same time and may be neglected. The relation then becomes $[{\bf T}_{n} = {\bf T}_{i}\alpha^{i}. \eqno(1.3.2.3) ]$ This relation is called the Cauchy relation, which allows the stress $[{\bf T}_{n}]$ to be expressed as a function of the stresses $[{\bf T}_{1}]$ , $[{\bf T}_{2}]$ and $[{\bf T}_{3}]$ that are applied to the three faces perpendicular to the axes, $[Px_{1}]$ , $[Px_{2}]$ and $[Px_{3}]$ . Let us project this relation onto these three axes: $[T_{nj} = T_{ij}\alpha_{i}. \eqno(1.3.2.4)]$ The nine components $[T_{ij}]$ are, by definition, the components of the stress tensor. In order to check that they are indeed the components of a tensor, it suffices to make the contracted product of each side of (1.3.2.4) by any vector $[x_{i}]$ : the left-hand side is a scalar product and the right-hand side a bilinear form. The $[T_{ij}]$ 's are therefore the components of a tensor. The index to the far left indicates the face to which the stress is applied (normal to the $[x_{1}]$ , $[x_{2} ]$ or $[x_{3}]$ axis), while the second one indicates on which axis the stress is projected.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 76-77